answer：Okay, so I have this problem where I need to find the value of ( a_8 ) in the expansion of ( (1+x)^{10} ) expressed as ( a_0 + a_1(1-x) + a_2(1-x)^2 + dots + a_{10}(1-x)^{10} ). Hmm, that seems a bit tricky, but let me try to figure it out step by step.First, I know that when you expand a binomial like ( (1+x)^{10} ), you can use the binomial theorem, which gives the coefficients as combinations. But in this case, instead of expanding in terms of ( x ), it's expanded in terms of ( (1 - x) ). So, I need to express ( (1+x)^{10} ) as a sum of powers of ( (1 - x) ) multiplied by coefficients ( a_0, a_1, dots, a_{10} ).I remember that sometimes when you have expressions like this, you can use substitution or generating functions to find the coefficients. Maybe I can let ( y = 1 - x ), so that ( x = 1 - y ). Then, substituting into the original expression, I get:( (1 + x)^{10} = (1 + (1 - y))^{10} = (2 - y)^{10} ).So, now the expression becomes ( (2 - y)^{10} ), which is equal to ( a_0 + a_1 y + a_2 y^2 + dots + a_{10} y^{10} ). Therefore, I need to expand ( (2 - y)^{10} ) and find the coefficient of ( y^8 ), which will be ( a_8 ).Alright, so let's recall the binomial expansion. The expansion of ( (a + b)^n ) is ( sum_{k=0}^{n} binom{n}{k} a^{n - k} b^k ). In this case, ( a = 2 ), ( b = -y ), and ( n = 10 ). So, each term in the expansion will be ( binom{10}{k} 2^{10 - k} (-y)^k ).Therefore, the coefficient of ( y^8 ) will be ( binom{10}{8} 2^{10 - 8} (-1)^8 ). Let me compute that.First, ( binom{10}{8} ) is the same as ( binom{10}{2} ) because ( binom{n}{k} = binom{n}{n - k} ). Calculating that, ( binom{10}{2} = frac{10 times 9}{2} = 45 ).Next, ( 2^{10 - 8} = 2^2 = 4 ).And ( (-1)^8 = 1 ) because any even power of -1 is positive.So, multiplying these together: ( 45 times 4 times 1 = 180 ).Therefore, the coefficient ( a_8 ) is 180.Wait, let me double-check my steps to make sure I didn't make a mistake. I substituted ( y = 1 - x ), which seems correct. Then, I expressed ( (1 + x)^{10} ) as ( (2 - y)^{10} ), which also seems right because ( 1 + x = 1 + (1 - y) = 2 - y ).Then, expanding ( (2 - y)^{10} ) using the binomial theorem, I correctly identified the general term as ( binom{10}{k} 2^{10 - k} (-y)^k ). So, the coefficient for ( y^8 ) is indeed ( binom{10}{8} 2^{2} (-1)^8 ).Calculating ( binom{10}{8} ) as 45 is correct because ( 10 times 9 / 2 = 45 ). Then, ( 2^2 = 4 ), and ( (-1)^8 = 1 ). Multiplying 45, 4, and 1 gives 180. So, I think my answer is correct.Alternatively, another way to approach this problem is by using generating functions or considering the transformation from ( x ) to ( 1 - x ). But since substitution worked smoothly, I think that method is solid.Just to be thorough, let me consider another approach. Suppose I don't substitute ( y = 1 - x ), but instead try to express ( (1 + x)^{10} ) in terms of ( (1 - x) ). That might involve expanding both sides and equating coefficients, but that seems more complicated.Alternatively, I could think of ( (1 + x) = 2 - (1 - x) ), so ( (1 + x)^{10} = (2 - (1 - x))^{10} ), which is the same as ( (2 - y)^{10} ) where ( y = 1 - x ). So, that's the same substitution as before, leading to the same conclusion.Therefore, I feel confident that ( a_8 = 180 ).**Final Answer**The value of ( a_8 ) is boxed{180}.

answer：Okay, so I have this problem here about converting polar equations to Cartesian and then finding the product of distances from a point to the intersection points of a line and a circle. Let me try to figure this out step by step.First, part (1) asks for the Cartesian equations of line l and circle C. Starting with circle C. The polar equation is given as ρ = 4 cos θ. I remember that in polar coordinates, ρ = 2a cos θ represents a circle with radius a, centered at (a, 0) in Cartesian coordinates. So, if ρ = 4 cos θ, that should be a circle with radius 2, centered at (2, 0). Let me verify that.In general, the conversion from polar to Cartesian coordinates is x = ρ cos θ and y = ρ sin θ. So, if I have ρ = 4 cos θ, I can multiply both sides by ρ to get ρ² = 4ρ cos θ. Then, substituting ρ² with x² + y² and ρ cos θ with x, I get x² + y² = 4x. To write this in standard form, I can rearrange it:x² - 4x + y² = 0.Completing the square for the x terms: take the coefficient of x, which is -4, divide by 2 to get -2, square it to get 4. So, add and subtract 4:(x² - 4x + 4) + y² = 4.Which simplifies to (x - 2)² + y² = 4. Yep, that's a circle centered at (2, 0) with radius 2. So that's the Cartesian equation of circle C.Now, moving on to the parametric equations of line l:x = (-3/5)t + 2y = (4/5)t + 1I need to convert this into a Cartesian equation. To do that, I can solve one of the equations for t and substitute into the other.Let's take the x equation:x = (-3/5)t + 2Subtract 2 from both sides:x - 2 = (-3/5)tMultiply both sides by (-5/3):t = (-5/3)(x - 2)Now, substitute this into the y equation:y = (4/5)t + 1Substitute t:y = (4/5)[(-5/3)(x - 2)] + 1Simplify:First, multiply (4/5) and (-5/3):(4/5)*(-5/3) = (-20)/15 = (-4)/3So, y = (-4/3)(x - 2) + 1Distribute the (-4/3):y = (-4/3)x + (8/3) + 1Convert 1 to thirds: 1 = 3/3So, y = (-4/3)x + (8/3 + 3/3) = (-4/3)x + 11/3So, the Cartesian equation of line l is y = (-4/3)x + 11/3.Alternatively, to write it in standard form (Ax + By + C = 0), I can multiply both sides by 3 to eliminate fractions:3y = -4x + 11Bring all terms to one side:4x + 3y - 11 = 0So, the standard form is 4x + 3y - 11 = 0.Alright, so part (1) is done. Circle C is (x - 2)² + y² = 4, and line l is 4x + 3y - 11 = 0 or y = (-4/3)x + 11/3.Now, part (2). We have point P(2,1). Line l intersects circle C at points A and B. We need to find |PA|·|PB|.Hmm, okay. So, I need to find points A and B where line l intersects circle C, then compute the distances from P to A and P to B, and then multiply those distances.Alternatively, I remember that in circle geometry, there's a theorem called the power of a point. The power of point P with respect to circle C is equal to |PA|·|PB|. So, maybe I can use that instead of finding A and B explicitly.Power of a point formula is |PA|·|PB| = |PO|² - r², where O is the center of the circle, and r is the radius. Wait, actually, no. Let me recall correctly.The power of a point P with respect to a circle with center O and radius r is defined as |PA|·|PB| = |PO|² - r², but only if P is outside the circle. If P is inside, then it's negative, but since we're taking absolute values, it might still work.Wait, actually, the power of a point P is equal to |PA|·|PB| regardless, but if P is inside the circle, it's negative, and if outside, positive. But since we're dealing with distances, maybe we take the absolute value.But let's verify.Given that P is (2,1), and the circle is centered at (2,0) with radius 2. So, the distance from P to the center O is sqrt[(2-2)² + (1 - 0)²] = sqrt[0 + 1] = 1. Since the radius is 2, and 1 < 2, point P is inside the circle. Therefore, the power of point P is negative, but since we're taking |PA|·|PB|, it should be equal to r² - |PO|².Wait, let me recall the exact formula.Power of a point P with respect to a circle is defined as:- If P is outside: |PA|·|PB| = |PO|² - r²- If P is inside: |PA|·|PB| = r² - |PO|²But actually, in both cases, the power is |PO|² - r², but when P is inside, it becomes negative, which is equal to -(|PA|·|PB|). So, depending on the convention, sometimes it's written as |PA|·|PB| = | |PO|² - r² |.But in our case, since we need |PA|·|PB|, regardless of the position, it's equal to | |PO|² - r² |. So, let's compute that.First, compute |PO|²: distance from P(2,1) to O(2,0):Distance squared is (2-2)² + (1 - 0)² = 0 + 1 = 1.Radius squared is 2² = 4.So, |PO|² - r² = 1 - 4 = -3. Taking absolute value, it's 3. Therefore, |PA|·|PB| = 3.Wait, that seems straightforward. So, maybe I don't need to find points A and B explicitly. But just to make sure, let me try solving it by finding A and B and computing the distances.Alternatively, I can parametrize the line l and substitute into the circle equation to find points A and B, then compute the distances.Let me try that method as a verification.So, parametric equations of line l are:x = (-3/5)t + 2y = (4/5)t + 1We can substitute these into the circle equation (x - 2)² + y² = 4.Substituting x and y:[ (-3/5 t + 2 - 2 ) ]² + [ (4/5 t + 1) ]² = 4Simplify:[ (-3/5 t ) ]² + [ (4/5 t + 1) ]² = 4Compute each term:First term: ( (-3/5 t) )² = (9/25)t²Second term: (4/5 t + 1)² = (16/25)t² + (8/5)t + 1So, adding both terms:9/25 t² + 16/25 t² + 8/5 t + 1 = 4Combine like terms:(9/25 + 16/25) t² + 8/5 t + 1 - 4 = 025/25 t² + 8/5 t - 3 = 0Simplify:t² + (8/5)t - 3 = 0Multiply through by 5 to eliminate fractions:5t² + 8t - 15 = 0Now, solve for t using quadratic formula:t = [ -8 ± sqrt(8² - 4*5*(-15)) ] / (2*5)Compute discriminant:64 + 300 = 364So,t = [ -8 ± sqrt(364) ] / 10Simplify sqrt(364):364 = 4*91 = 4*13*7, so sqrt(364) = 2*sqrt(91)Therefore,t = [ -8 ± 2sqrt(91) ] / 10 = [ -4 ± sqrt(91) ] / 5So, the two values of t are t1 = [ -4 + sqrt(91) ] / 5 and t2 = [ -4 - sqrt(91) ] / 5.Now, find points A and B by plugging these t values back into parametric equations.First, for t1:x1 = (-3/5)t1 + 2 = (-3/5)( [ -4 + sqrt(91) ] / 5 ) + 2Compute:= (12 - 3sqrt(91))/25 + 2= (12 - 3sqrt(91) + 50)/25= (62 - 3sqrt(91))/25Similarly, y1 = (4/5)t1 + 1 = (4/5)( [ -4 + sqrt(91) ] / 5 ) + 1= ( -16 + 4sqrt(91) ) / 25 + 1= ( -16 + 4sqrt(91) + 25 ) / 25= (9 + 4sqrt(91))/25So, point A is ( (62 - 3sqrt(91))/25 , (9 + 4sqrt(91))/25 )Similarly, for t2:x2 = (-3/5)t2 + 2 = (-3/5)( [ -4 - sqrt(91) ] / 5 ) + 2= (12 + 3sqrt(91))/25 + 2= (12 + 3sqrt(91) + 50)/25= (62 + 3sqrt(91))/25y2 = (4/5)t2 + 1 = (4/5)( [ -4 - sqrt(91) ] / 5 ) + 1= ( -16 - 4sqrt(91) ) / 25 + 1= ( -16 - 4sqrt(91) + 25 ) / 25= (9 - 4sqrt(91))/25So, point B is ( (62 + 3sqrt(91))/25 , (9 - 4sqrt(91))/25 )Now, we have points A and B. Point P is (2,1). Let's compute |PA| and |PB|.First, compute |PA|:PA is the distance between P(2,1) and A( (62 - 3sqrt(91))/25 , (9 + 4sqrt(91))/25 )Compute the differences:Δx = 2 - (62 - 3sqrt(91))/25 = (50 - 62 + 3sqrt(91))/25 = (-12 + 3sqrt(91))/25Δy = 1 - (9 + 4sqrt(91))/25 = (25 - 9 - 4sqrt(91))/25 = (16 - 4sqrt(91))/25So, |PA|² = [ (-12 + 3sqrt(91))/25 ]² + [ (16 - 4sqrt(91))/25 ]²Compute each term:First term: [ (-12 + 3sqrt(91)) ]² = (144 - 72sqrt(91) + 9*91) = 144 - 72sqrt(91) + 819 = 963 - 72sqrt(91)Divide by 25²=625: (963 - 72sqrt(91))/625Second term: [ (16 - 4sqrt(91)) ]² = (256 - 128sqrt(91) + 16*91) = 256 - 128sqrt(91) + 1456 = 1712 - 128sqrt(91)Divide by 625: (1712 - 128sqrt(91))/625Add both terms:(963 - 72sqrt(91) + 1712 - 128sqrt(91))/625 = (2675 - 200sqrt(91))/625Simplify:2675 / 625 = 4.28, but let's see: 625*4 = 2500, so 2675 - 2500 = 175, so 4 + 175/625 = 4 + 7/25 = 4.28Similarly, 200sqrt(91)/625 = (200/625)sqrt(91) = (4/12.5)sqrt(91) = (8/25)sqrt(91)Wait, 200/625 simplifies to 8/25.So, |PA|² = (2675 - 200sqrt(91))/625 = (2675/625) - (200sqrt(91)/625) = 4.28 - (8/25)sqrt(91)Wait, but 2675 divided by 625: 625*4=2500, 2675-2500=175, so 175/625=7/25. So, 2675/625=4 + 7/25=4.28.Similarly, 200/625=8/25.So, |PA|² = 4 + 7/25 - (8/25)sqrt(91) = (100/25 + 7/25) - (8/25)sqrt(91) = (107/25) - (8/25)sqrt(91)Similarly, compute |PB|²:Point B is ( (62 + 3sqrt(91))/25 , (9 - 4sqrt(91))/25 )Compute Δx = 2 - (62 + 3sqrt(91))/25 = (50 - 62 - 3sqrt(91))/25 = (-12 - 3sqrt(91))/25Δy = 1 - (9 - 4sqrt(91))/25 = (25 - 9 + 4sqrt(91))/25 = (16 + 4sqrt(91))/25So, |PB|² = [ (-12 - 3sqrt(91))/25 ]² + [ (16 + 4sqrt(91))/25 ]²Compute each term:First term: [ (-12 - 3sqrt(91)) ]² = (144 + 72sqrt(91) + 9*91) = 144 + 72sqrt(91) + 819 = 963 + 72sqrt(91)Divide by 625: (963 + 72sqrt(91))/625Second term: [ (16 + 4sqrt(91)) ]² = (256 + 128sqrt(91) + 16*91) = 256 + 128sqrt(91) + 1456 = 1712 + 128sqrt(91)Divide by 625: (1712 + 128sqrt(91))/625Add both terms:(963 + 72sqrt(91) + 1712 + 128sqrt(91))/625 = (2675 + 200sqrt(91))/625Simplify:2675/625 = 4.28 as before, and 200sqrt(91)/625 = (8/25)sqrt(91)So, |PB|² = (2675 + 200sqrt(91))/625 = 4.28 + (8/25)sqrt(91) = (107/25) + (8/25)sqrt(91)Now, compute |PA|·|PB|.Note that |PA|²·|PB|² = (|PA|·|PB|)²So, (|PA|·|PB|)² = |PA|²·|PB|² = [ (107/25 - 8sqrt(91)/25 ) ] · [ (107/25 + 8sqrt(91)/25 ) ]This is a difference of squares: (a - b)(a + b) = a² - b²So, compute:(107/25)² - (8sqrt(91)/25)²Compute each term:(107/25)² = (107²)/(25²) = 11449/625(8sqrt(91)/25)² = (64*91)/625 = (5824)/625So,(11449 - 5824)/625 = (5625)/625 = 9Therefore, (|PA|·|PB|)² = 9, so |PA|·|PB| = sqrt(9) = 3.So, that's consistent with the power of a point result.Therefore, the value is 3.Alternatively, since I got the same answer both ways, I can be confident that |PA|·|PB| = 3.**Final Answer**The value of (|PA|cdot|PB|) is boxed{3}).

answer：Okay, so I have this sequence problem here. It says that the sequence ( a_n ) is defined by ( a_1 = 1 ), and for each ( n geq 1 ), the next term is given by ( a_{n+1} = a_n + frac{2 a_n}{n} ). I need to find ( a_{200} ). Hmm, let me try to figure this out step by step.First, let me write down the recurrence relation again to make sure I have it correctly:[ a_{n+1} = a_n + frac{2 a_n}{n} ]Hmm, so each term is the previous term plus twice the previous term divided by ( n ). Maybe I can factor out ( a_n ) to simplify this expression. Let me try that:[ a_{n+1} = a_n left(1 + frac{2}{n}right) ]Yes, that looks better. So, ( a_{n+1} ) is equal to ( a_n ) multiplied by ( left(1 + frac{2}{n}right) ). This seems like a multiplicative recurrence relation, which might mean that I can express ( a_n ) as a product of these factors.Let me try to write out the first few terms to see if I can spot a pattern.Starting with ( a_1 = 1 ).Then, ( a_2 = a_1 left(1 + frac{2}{1}right) = 1 times 3 = 3 ).Next, ( a_3 = a_2 left(1 + frac{2}{2}right) = 3 times 2 = 6 ).Then, ( a_4 = a_3 left(1 + frac{2}{3}right) = 6 times frac{5}{3} = 10 ).Wait, ( 6 times frac{5}{3} ) is ( 10 ). Okay, moving on.( a_5 = a_4 left(1 + frac{2}{4}right) = 10 times frac{3}{2} = 15 ).Hmm, ( a_6 = a_5 left(1 + frac{2}{5}right) = 15 times frac{7}{5} = 21 ).Wait a second, these numbers look familiar. 1, 3, 6, 10, 15, 21... These are triangular numbers! The nth triangular number is given by ( frac{n(n+1)}{2} ). Let me check if that's the case here.For ( n = 1 ): ( frac{1 times 2}{2} = 1 ). Correct.For ( n = 2 ): ( frac{2 times 3}{2} = 3 ). Correct.For ( n = 3 ): ( frac{3 times 4}{2} = 6 ). Correct.For ( n = 4 ): ( frac{4 times 5}{2} = 10 ). Correct.For ( n = 5 ): ( frac{5 times 6}{2} = 15 ). Correct.For ( n = 6 ): ( frac{6 times 7}{2} = 21 ). Correct.So, it seems that ( a_n = frac{n(n+1)}{2} ). If that's the case, then ( a_{200} ) would be ( frac{200 times 201}{2} ). Let me compute that.But before I jump to conclusions, I should verify if this formula actually satisfies the given recurrence relation. Let's test it.Assume ( a_n = frac{n(n+1)}{2} ). Then, ( a_{n+1} ) should be ( a_n left(1 + frac{2}{n}right) ).Compute ( a_{n+1} ) using the formula:[ a_{n+1} = frac{(n+1)(n+2)}{2} ]Now, compute ( a_n left(1 + frac{2}{n}right) ):[ a_n left(1 + frac{2}{n}right) = frac{n(n+1)}{2} times left(1 + frac{2}{n}right) ]Simplify the expression inside the parentheses:[ 1 + frac{2}{n} = frac{n + 2}{n} ]So, multiplying:[ frac{n(n+1)}{2} times frac{n + 2}{n} = frac{(n+1)(n + 2)}{2} ]Which is exactly ( a_{n+1} ). So, yes, the formula ( a_n = frac{n(n+1)}{2} ) satisfies the recurrence relation. Therefore, the general term is indeed the nth triangular number.Therefore, ( a_{200} = frac{200 times 201}{2} ).Let me compute that:First, compute ( 200 times 201 ). 200 times 200 is 40,000, and 200 times 1 is 200, so total is 40,200.Then, divide by 2: 40,200 / 2 = 20,100.So, ( a_{200} = 20,100 ).Wait, let me double-check my calculations to be sure.200 multiplied by 201: 200 * 200 = 40,000; 200 * 1 = 200; so 40,000 + 200 = 40,200. Correct.Divide by 2: 40,200 / 2 = 20,100. Correct.Alternatively, I can compute 200 * 201 / 2 as (200 / 2) * 201 = 100 * 201 = 20,100. Same result.So, I think that's solid. Therefore, the answer is 20,100.But just to be thorough, let me check the initial terms again with the formula.For ( n = 1 ): 1*2/2 = 1. Correct.For ( n = 2 ): 2*3/2 = 3. Correct.For ( n = 3 ): 3*4/2 = 6. Correct.For ( n = 4 ): 4*5/2 = 10. Correct.So, the formula holds for the initial terms, and it satisfies the recurrence relation. Therefore, I can confidently say that ( a_n = frac{n(n+1)}{2} ), so ( a_{200} = 20,100 ).**Final Answer**The value of ( a_{200} ) is boxed{20100}.

answer：To solve the first expression, I'll add the numbers sequentially.For the second expression, I'll first calculate the multiplication before adding the remaining numbers.In the third expression, I'll group the numbers to simplify the addition.

answer：To determine how many numbers Grace writes are positive, I'll start by understanding the sequence she creates.The first number is 43, and each subsequent number decreases by 4. This means the sequence is: 43, 39, 35, 31, ..., and so on.I can represent this sequence using the formula for the nth term of an arithmetic sequence:aₙ = a₁ + (n - 1)dwhere a₁ is the first term (43), d is the common difference (-4), and n is the term number.Plugging in the values:aₙ = 43 + (n - 1)(-4) = 43 - 4n + 4 = 47 - 4nTo find the last positive number, I'll set aₙ greater than 0:47 - 4n > 047 > 4nn < 47/4n < 11.75Since n must be an integer, the largest possible value for n is 11. This means the 11th term is the last positive number in the sequence.Therefore, Grace writes 11 positive numbers.

answer：Okay, so I need to calculate the length of a cycloid given by the parametric equations ( x = a(t - sin t) ) and ( y = a(1 - cos t) ) where ( t ) ranges from 0 to ( 2pi ). Hmm, I remember that the formula for the length of a parametric curve is an integral involving the derivatives of ( x ) and ( y ) with respect to the parameter ( t ). Let me try to recall the exact formula.I think it's something like the integral from ( t = 0 ) to ( t = 2pi ) of the square root of ( (dx/dt)^2 + (dy/dt)^2 ) dt. Yeah, that sounds right. So, first, I need to find the derivatives ( dx/dt ) and ( dy/dt ).Starting with ( x = a(t - sin t) ). Taking the derivative with respect to ( t ), that should be ( dx/dt = a(1 - cos t) ). Let me double-check that. The derivative of ( t ) is 1, and the derivative of ( sin t ) is ( cos t ), so yes, ( dx/dt = a(1 - cos t) ).Now for ( y = a(1 - cos t) ). The derivative ( dy/dt ) would be ( a(0 + sin t) ) because the derivative of ( cos t ) is ( -sin t ), so the negative cancels out. So, ( dy/dt = a sin t ).Okay, so now I have both derivatives. Next step is to square each of them and add them together. Let me compute ( (dx/dt)^2 + (dy/dt)^2 ).First, ( (dx/dt)^2 = [a(1 - cos t)]^2 = a^2 (1 - 2cos t + cos^2 t) ).Second, ( (dy/dt)^2 = [a sin t]^2 = a^2 sin^2 t ).Adding these together: ( a^2 (1 - 2cos t + cos^2 t) + a^2 sin^2 t ).Let me factor out the ( a^2 ) to make it simpler: ( a^2 [1 - 2cos t + cos^2 t + sin^2 t] ).I remember that ( cos^2 t + sin^2 t = 1 ), so substituting that in, we get ( a^2 [1 - 2cos t + 1] ).Simplifying inside the brackets: ( 1 + 1 = 2 ), so it becomes ( a^2 [2 - 2cos t] ).Factor out a 2: ( a^2 times 2(1 - cos t) ) which is ( 2a^2 (1 - cos t) ).So, the integrand becomes the square root of ( 2a^2 (1 - cos t) ). Let me write that as ( sqrt{2a^2 (1 - cos t)} ).Simplify the square root: ( sqrt{2a^2} times sqrt{1 - cos t} ). Since ( a ) is a positive constant, ( sqrt{a^2} = a ), so this becomes ( a sqrt{2} times sqrt{1 - cos t} ).Hmm, so the integral for the length is ( int_{0}^{2pi} a sqrt{2} sqrt{1 - cos t} , dt ).I can factor out the constants ( a sqrt{2} ) from the integral, so it becomes ( a sqrt{2} int_{0}^{2pi} sqrt{1 - cos t} , dt ).Now, I need to evaluate this integral ( int_{0}^{2pi} sqrt{1 - cos t} , dt ). This seems a bit tricky. Maybe I can use a trigonometric identity to simplify the expression inside the square root.I remember that ( 1 - cos t ) can be expressed using a double-angle identity. Let me recall: ( 1 - cos t = 2 sin^2(t/2) ). Yes, that's correct because ( cos t = 1 - 2 sin^2(t/2) ), so rearranging gives ( 1 - cos t = 2 sin^2(t/2) ).So, substituting that into the integral, ( sqrt{1 - cos t} = sqrt{2 sin^2(t/2)} ). The square root of ( sin^2(t/2) ) is ( |sin(t/2)| ). Since ( t ) is going from 0 to ( 2pi ), ( t/2 ) goes from 0 to ( pi ), and ( sin(t/2) ) is non-negative in this interval. So, we can drop the absolute value and write it as ( sqrt{2} sin(t/2) ).Therefore, the integral becomes ( sqrt{2} int_{0}^{2pi} sin(t/2) , dt ).Wait, hold on, let me make sure I did that correctly. So, ( sqrt{1 - cos t} = sqrt{2 sin^2(t/2)} = sqrt{2} |sin(t/2)| ). Since ( t ) is between 0 and ( 2pi ), ( t/2 ) is between 0 and ( pi ), where ( sin(t/2) ) is indeed non-negative. So, yes, it's ( sqrt{2} sin(t/2) ).So, substituting back into the integral, we have:( a sqrt{2} times sqrt{2} int_{0}^{2pi} sin(t/2) , dt ).Wait, wait, hold on. Let me retrace. The integrand after substitution is ( sqrt{2} sin(t/2) ), so the integral is ( a sqrt{2} times sqrt{2} int_{0}^{2pi} sin(t/2) , dt ).Wait, that seems off. Let me go back step by step.We had:( sqrt{1 - cos t} = sqrt{2} sin(t/2) ).So, the integral ( int_{0}^{2pi} sqrt{1 - cos t} , dt = int_{0}^{2pi} sqrt{2} sin(t/2) , dt ).So, that's ( sqrt{2} int_{0}^{2pi} sin(t/2) , dt ).So, the entire expression for the length is ( a sqrt{2} times sqrt{2} int_{0}^{2pi} sin(t/2) , dt ).Wait, no, hold on. The original integral was ( a sqrt{2} times int sqrt{1 - cos t} dt ), which became ( a sqrt{2} times sqrt{2} int sin(t/2) dt ).So, ( sqrt{2} times sqrt{2} = 2 ), so the expression becomes ( 2a int_{0}^{2pi} sin(t/2) , dt ).Okay, that seems better. So, the integral simplifies to ( 2a int_{0}^{2pi} sin(t/2) , dt ).Now, I need to compute ( int_{0}^{2pi} sin(t/2) , dt ).Let me make a substitution to evaluate this integral. Let ( u = t/2 ), so ( du = dt/2 ), which means ( dt = 2 du ). When ( t = 0 ), ( u = 0 ), and when ( t = 2pi ), ( u = pi ).Substituting, the integral becomes ( int_{0}^{pi} sin u times 2 du ) which is ( 2 int_{0}^{pi} sin u , du ).The integral of ( sin u ) is ( -cos u ), so evaluating from 0 to ( pi ):( 2 [ -cos pi + cos 0 ] = 2 [ -(-1) + 1 ] = 2 [1 + 1] = 2 times 2 = 4 ).So, the integral ( int_{0}^{2pi} sin(t/2) , dt = 4 ).Therefore, going back to the expression for the length, it's ( 2a times 4 = 8a ).Wait, hold on, that can't be right because I remember that the length of one arch of a cycloid is ( 8a ). Hmm, but wait, let me verify.Wait, so if the integral is 4, then 2a times 4 is 8a. So, yes, that's correct. So, the length of the cycloid from ( t = 0 ) to ( t = 2pi ) is ( 8a ).But just to make sure I didn't make any mistakes in my calculations, let me go through the steps again.1. Start with parametric equations ( x = a(t - sin t) ) and ( y = a(1 - cos t) ).2. Compute derivatives: ( dx/dt = a(1 - cos t) ), ( dy/dt = a sin t ).3. Square and add: ( (a(1 - cos t))^2 + (a sin t)^2 = a^2(1 - 2cos t + cos^2 t + sin^2 t) ).4. Simplify using ( cos^2 t + sin^2 t = 1 ): ( a^2(2 - 2cos t) = 2a^2(1 - cos t) ).5. Square root: ( sqrt{2a^2(1 - cos t)} = a sqrt{2(1 - cos t)} ).6. Use identity ( 1 - cos t = 2 sin^2(t/2) ): ( a sqrt{2 times 2 sin^2(t/2)} = a sqrt{4 sin^2(t/2)} = a times 2 |sin(t/2)| ).7. Since ( t ) is between 0 and ( 2pi ), ( sin(t/2) ) is non-negative, so it's ( 2a sin(t/2) ).8. Therefore, the integrand becomes ( 2a sin(t/2) ).9. The integral for the length is ( int_{0}^{2pi} 2a sin(t/2) , dt ).10. Factor out the 2a: ( 2a int_{0}^{2pi} sin(t/2) , dt ).11. Substitute ( u = t/2 ), ( du = dt/2 ), so ( dt = 2 du ), limits from 0 to ( pi ).12. Integral becomes ( 2a times 2 int_{0}^{pi} sin u , du = 4a int_{0}^{pi} sin u , du ).13. Integrate ( sin u ): ( -cos u ) evaluated from 0 to ( pi ) is ( -cos pi + cos 0 = -(-1) + 1 = 2 ).14. So, the integral is ( 4a times 2 = 8a ).Wait, hold on, that seems conflicting with my earlier step. Wait, in step 10, I had ( 2a times int sin(t/2) dt ), which became ( 2a times 2 int sin u du ) which is ( 4a times int sin u du ). Then the integral of ( sin u ) from 0 to ( pi ) is 2, so 4a times 2 is 8a.Wait, but in my initial calculation, I thought the integral of ( sin(t/2) ) from 0 to ( 2pi ) was 4, but when I did substitution, I got 4a times 2, which is 8a. Hmm, perhaps I made a miscalculation earlier.Wait, let me clarify:After substitution, the integral ( int_{0}^{2pi} sin(t/2) dt ) becomes ( 2 int_{0}^{pi} sin u du ).Compute ( int_{0}^{pi} sin u du = [-cos u]_0^{pi} = (-cos pi) - (-cos 0) = (-(-1)) - (-1) = 1 + 1 = 2 ).So, ( 2 times 2 = 4 ). Therefore, ( int_{0}^{2pi} sin(t/2) dt = 4 ).Therefore, going back, the length is ( 2a times 4 = 8a ). So, that's correct.Wait, but earlier, when I thought I had 2a times 4, that was correct because the integral was 4, so 2a times 4 is 8a. So, yes, the length is 8a.I think I confused myself earlier when I thought I had 2a times 4, but actually, the substitution led to 4, so 2a times 4 is 8a.So, to recap:- The parametric equations are given.- Compute derivatives, square and add, simplify using trig identities.- The integrand simplifies to ( 2a sin(t/2) ).- The integral over 0 to ( 2pi ) is 4.- Multiply by 2a to get 8a.Therefore, the length of the cycloid is ( 8a ).I think that's correct. Let me just check with another method or recall if the standard length of a cycloid is indeed ( 8a ). Yes, I remember that the length of one arch of a cycloid is ( 8a ), so that confirms it.So, I think I've done everything correctly, and the final answer is ( 8a ).**Final Answer**The length of the cycloid is boxed{8a}.